August 12, 2019 - August 16, 2019
Public Access Category: All |
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- Time:
- 2:00pm-3:30pm
- Location:
- Room No. 215 Department of Mathematics
- Description:
- Commutative Algebra Seminars
Speaker: Dilip Patil.
Affiliation: IISc, Bangalore.
Seminar I - Date and Time: Tuesday 13 August, 2:00 pm - 3:30 pm.
Venue: Room 215, Department of Mathematics.
Title: The Cohen-Structure Theorems.
Abstract: The purpose of these two lectures is to provide the proof of
Cohen’s structure theorem for complete local rings (which Cohen proved in
his PhD thesis 1942, Johns Hopkins University under the guidance of Oscar
Zariski). In these lecture we deal with the equicharacteristic case. We
give a modern and concise treatment by using the notion of formal
smoothness which was introduced by Grothendieck in 1964 in EGA Chapter IV.
It is closely connected with the differentials and throws new light to the
theory of regular local rings and used in proving Cohen’s structure
theorem of complete local rings.
- Time:
- 4:00pm-5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Algebraic groups seminar.
Speaker: Uday Bhaskar.
Affiliation: TIFR, Mumbai.
Date and Time: Tuesday 13 August, 4:00 pm - 5:00 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Simultaneous conjugacy classes of commuting tuples of matrices.
Abstract: We discuss the classification of tuples of commuting matrices
over a finite field, up to simultaneous conjugation.
- Time:
- 4:00pm-5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Mathematics Colloquium.
Speaker: Saikat Mazumdar.
Affiliation: IIT Bombay.
Date and Time: Wednesday 14 August, 4:00 pm - 5:00 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Yamabe problem and beyond: an interplay of geometry and PDE.
Abstract: Motivated by the theory of compact surfaces, Yamabe wanted to
show that on a given compact Riemannian manifold of any dimension there
always exists a (conformal) metric with constant scalar curvature. It
turns out that solving the Yamabe problem amounts to solving a nonlinear
elliptic partial differential equation (PDE). The solution of the Yamabe
problem by Trudinger, Aubin and Schoen highlighted the local and global
nature of the problem and the unexpected role of the positive mass theorem
of general relativity. In the first part of my talk, I will survey the
Yamabe problem and the related issues of the compactness of solutions.
In the second part of the talk, I will discuss the higher-order or
polyharmonic version of the Yamabe problem: "Given a compact Riemannian
manifold (M, g), does there exists a metric conformal to g with constant
Q-curvature?" The behaviour of Q-curvature under conformal changes of the
metric is governed by certain conformally covariant powers of the
Laplacian. The problem of prescribing the Q-curvature in a conformal class
then amounts to solving a nonlinear elliptic PDE involving the powers of
Laplacian called the GJMS operator. In general the explicit form of this
GJMS operator is unknown. This together with a lack of maximum principle
makes the problem difficult to tackle. I will present some of my results
in this direction and mention some recent progress.
- Time:
- 2:00pm-3:30pm
- Location:
- Room No. 215 Department of Mathematics
- Description:
- Commutative Algebra Seminars
Speaker: Dilip Patil.
Affiliation: IISc, Bangalore.
Seminar II - Date and Time: Friday 16 August, 2:00 pm - 3:30 pm.
Venue: Room 215, Department of Mathematics.
Title: The Cohen-Structure Theorems.
Abstract: The purpose of these two lectures is to provide the proof of
Cohen’s structure theorem for complete local rings (which Cohen proved in
his PhD thesis 1942, Johns Hopkins University under the guidance of Oscar
Zariski). In these lecture we deal with the equicharacteristic case. We
give a modern and concise treatment by using the notion of formal
smoothness which was introduced by Grothendieck in 1964 in EGA Chapter IV.
It is closely connected with the differentials and throws new light to the
theory of regular local rings and used in proving Cohen’s structure
theorem of complete local rings.
- Time:
- 4:00pm-5:00pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Speaker: Sampat Kumar Sharma.
Affiliation: ISI, Kolkata.
Date and Time: Friday 16 August, 4:00 pm - 5:00 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: On a question of Suslin about completion of unimodular rows.
Abstract:
R.G. Swan and J. Towber showed that if (a
2
, b, c) is a unimodular row
over any commutative ring R then it can be completed to an invertible
matrix over R. This was strikingly generalised by A.A. Suslin who showed
that if (a
r!
0 , a1, . . . , ar) is a unimodular row over R then it can be com-
pleted to an invertible matrix. As a consequence A.A. Suslin proceeds to
conclude that if 1
r! ∈ R, then a unimodular row v(X) ∈ Umr+1(R[X])
of degree one, with v(0) = (1, 0, . . . , 0), is completable to an invertible
matrix. Then he asked
(Sr(R)): Let R be a local ring such that r! ∈ GL1(R), and let p =
(f0(X), . . . , fr(X)) ∈ Umr+1(R[X]) with p(0) = e1(= (1, 0, . . . , 0)). Is it
possible to embed the row p in an invertible matrix?
Due to Suslin, one knows answer to this question when r = d + 1,
without the assumption r! ∈ GL1(R). In 1988, Ravi Rao answered this
question in the case when r = d.
In this talk we will discuss about the Suslin’s question Sr(R) when r =
d − 1. We will also discuss about two important ingredients; “homotopy
and commutativity principle” and “absence of torsion in Umd+1(R[X])
Ed+1(R[X]) ”,
to answer Suslin’s question in the case when r = d − 1, where d is the
dimension of the ring.